There is now a Park & Ride service to and from the UTMB Galveston area. "UTMB commuters will have a new transit choice starting April 30 with the introduction of Island Connect, the first park-and-ride service between Galveston and the Clear Lake-League City area."
It is actually A Van Pool or a Car Pool. If you google Van Pool or Car Pool for Galveston, you will find multiples. I don't want to do it for you, as I want you to see how many that there are and it is a safe bet. People just meet at I-45 in some areas, or a shopping center at a certain time and they drop you off at a certain time.
If you are working at UTMB there is a Van Share that will take you back and forth. You need to check into that. Sometimes there are other van shares as well, but that is probably the most inexpensive and easiest, besides you driving yourself.